Exponential EquationsDefinition, Workings, and Examples
In math, an exponential equation occurs when the variable appears in the exponential function. This can be a terrifying topic for kids, but with a some of instruction and practice, exponential equations can be solved quickly.
This blog post will discuss the definition of exponential equations, types of exponential equations, proceduce to figure out exponential equations, and examples with answers. Let's get right to it!
What Is an Exponential Equation?
The first step to solving an exponential equation is determining when you are working with one.
Definition
Exponential equations are equations that have the variable in an exponent. For instance, 2x+1=0 is not an exponential equation, but 2x+1=0 is an exponential equation.
There are two primary things to keep in mind for when you seek to determine if an equation is exponential:
1. The variable is in an exponent (signifying it is raised to a power)
2. There is only one term that has the variable in it (in addition of the exponent)
For example, check out this equation:
y = 3x2 + 7
The first thing you should observe is that the variable, x, is in an exponent. The second thing you should observe is that there is another term, 3x2, that has the variable in it – just not in an exponent. This implies that this equation is NOT exponential.
On the flipside, look at this equation:
y = 2x + 5
Once again, the first thing you must note is that the variable, x, is an exponent. The second thing you must note is that there are no more terms that have the variable in them. This implies that this equation IS exponential.
You will come upon exponential equations when you try solving different calculations in compound interest, algebra, exponential growth or decay, and various distinct functions.
Exponential equations are essential in math and play a pivotal responsibility in solving many mathematical problems. Thus, it is important to fully understand what exponential equations are and how they can be utilized as you move ahead in your math studies.
Types of Exponential Equations
Variables appear in the exponent of an exponential equation. Exponential equations are surprisingly common in everyday life. There are three major kinds of exponential equations that we can solve:
1) Equations with the same bases on both sides. This is the simplest to solve, as we can easily set the two equations equal to each other and solve for the unknown variable.
2) Equations with dissimilar bases on both sides, but they can be created similar utilizing rules of the exponents. We will put a few examples below, but by converting the bases the equal, you can follow the exact steps as the first case.
3) Equations with variable bases on each sides that is impossible to be made the same. These are the trickiest to solve, but it’s possible through the property of the product rule. By raising two or more factors to identical power, we can multiply the factors on each side and raise them.
Once we have done this, we can resolute the two new equations equal to each other and figure out the unknown variable. This blog does not contain logarithm solutions, but we will tell you where to get guidance at the very last of this blog.
How to Solve Exponential Equations
After going through the definition and kinds of exponential equations, we can now understand how to solve any equation by following these simple steps.
Steps for Solving Exponential Equations
Remember these three steps that we are required to follow to solve exponential equations.
Primarily, we must determine the base and exponent variables in the equation.
Second, we have to rewrite an exponential equation, so all terms are in common base. Then, we can work on them through standard algebraic techniques.
Lastly, we have to work on the unknown variable. Now that we have figured out the variable, we can put this value back into our first equation to find the value of the other.
Examples of How to Solve Exponential Equations
Let's look at a few examples to note how these procedures work in practice.
First, we will solve the following example:
7y + 1 = 73y
We can see that all the bases are the same. Hence, all you are required to do is to restate the exponents and solve using algebra:
y+1=3y
y=½
Right away, we replace the value of y in the specified equation to corroborate that the form is true:
71/2 + 1 = 73(½)
73/2=73/2
Let's follow this up with a further complicated question. Let's solve this expression:
256=4x−5
As you can see, the sides of the equation do not share a identical base. Despite that, both sides are powers of two. In essence, the solution includes decomposing respectively the 4 and the 256, and we can substitute the terms as follows:
28=22(x-5)
Now we work on this expression to find the ultimate answer:
28=22x-10
Perform algebra to solve for x in the exponents as we conducted in the last example.
8=2x-10
x=9
We can verify our answer by replacing 9 for x in the initial equation.
256=49−5=44
Continue searching for examples and problems over the internet, and if you utilize the properties of exponents, you will inturn master of these theorems, solving almost all exponential equations with no issue at all.
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Solving problems with exponential equations can be tough in absence help. Although this guide take you through the essentials, you still might encounter questions or word questions that might stumble you. Or perhaps you require some further guidance as logarithms come into the scene.
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